I have not heard or used this phrase before, but can I use the sentence "please mute your voice" in conversation?
Since in the current electronic age mute has come to mean to silence completely (not merely to soften, as in the technical musical direction con sordino), the actual meaning of the sentence you propose comes awfully close to STFU, so it perhaps depends on the strength of your jaw (to withstand the blows of your interlocutor’s fist). To speak plainly, the sentence you propose is likely to prove highly offensive. “Please lower your voice” is much safer for your jaw.
In a face-to-face conversation, "mute your voice" would be awkward. But in the context of an online meeting, where people from different places join in over a common conferencing bridge, the speaker could politely request others to "go on mute" (to mute their mics for the time being) so as to avoid unnecessary noise over the bridge.
The phrase "Please mute your voice." bears with it a subtle but somewhat unsavoury mix of modernity and arrogance. It is similar to saying "Kindly still your tongue."
Adding "please" within the sentence is a little like adding sugar to a bitter brew, and is more effective when preceding the bitter part of the sentence (Please mute your voice) rather than following it (Mute your voice please), at the later scenario is more likely to be taken as a hasty addition.
If the intention is to get somebody to be quiet without being rude about it (and presuming that this is the first attempt) then it can be more effective to invoke the element of indirect peer pressure to get the desired result.
"Kindly note that it is important that we have absolute silence so that the rest of us can better..." (enjoy the show/ follow the lecture/ relax)
If brevity is a requirement then the more assertive "Silence please." coupled with a deliberately unfocused physical mannerism (i.e. not focusing upon the offender) might work well enough.
I have almost always heard the expression "mute your voice" in teleconference calls, and only rarely in person.
99% of the uses were on teleconferences, where 'mute' meant "disaable using the mute feature on your phone or the teleconference software" and 'your voice' really meant "your connection". There was almost never a negative implication if it was just a noisy line, but if the 'loud voice' was engaged in a side conversation without muting, the request often had a negative tone. 'Please' was optional.
The other 1%, during in-person conversations, were definitely insulting, or at least could be heard that way, assuming it wasn't a needling joke. 'Please' was never included.
So, never use it in person unless as a joke with close friends. On telecons, try not to be annoyed or sound annoyed when you say it, whether you use please or not.
(I've never heard 'mute your voice' mean 'lower your voice', I don't think it is ever used that way.)
The verb form of mute is usually defined as "to make quiet" or "to silence", which carries a slight implication that the person doing the muting isn't the one making the noise in the first place - instead of making noise and then muting it, it would make sense for me to simply not make the noise in the first place. When you mute a stereo or television, you're canceling the noise made independently by the device.
This isn't an explicit part of the definition, but is a connotation that I feel is there, and that is why "mute your voice!" sounds strange to me, whereas "lower your voice" or "don't make so much noise" sounds a lot better - you're asking someone to modulate the noise they're making, not to actively silence an unruly and uncontrolled noisemaker.
I think it is the meaning of to mute used on a figurative sense:
(Instruments) to reduce the volume of (a musical instrument) by means of a mute, soft pedal, etc
Speak lower please is a more appropriate expression.